1. Uninformed search strategies A search strategy is defined by picking the order of node expansion Uninformed search strategies use only the information available in the problem definition – Breadth-first search – Depth-first search – Iterative deepening search – Uniform-cost search

2. Breadth-first search Expand shallowest unexpanded node Implementation: frontier is a FIFO queue Example state space graph for a tiny search problem Example from P. Abbeel and D. Klein

3. Breadth-first search Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)

4. Depth-first search Expand deepest unexpanded node Implementation: frontier is a LIFO queue

5. Depth-first search Expansion order: (d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)

6. http://xkcd.com/761/

7. Analysis of search strategies Strategies are evaluated along the following criteria: – Completeness: does it always find a solution if one exists? – Optimality: does it always find a least-cost solution? – Time complexity: number of nodes generated – Space complexity: maximum number of nodes in memory Time and space complexity are measured in terms of – b: maximum branching factor of the search tree – d: depth of the optimal solution – m: maximum length of any path in the state space (may be infinite)

8. Properties of breadth-first search Complete? Yes (if branching factor b is finite) Optimal? Yes – if cost = 1 per step Time? Number of nodes in a b-ary tree of depth d: O(b d ) (d is the depth of the optimal solution) Space? O(bd)O(bd) Space is the bigger problem (more than time)

9. Properties of depth-first search Complete? Fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path  complete in finite spaces Optimal? No – returns the first solution it finds Time? Could be the time to reach a solution at maximum depth m: O(b m ) Terrible if m is much larger than d But if there are lots of solutions, may be much faster than BFS Space? O(bm), i.e., linear space!

10. Iterative deepening search Use DFS as a subroutine 1.Check the root 2.Do a DFS searching for a path of length 1 3.If there is no path of length 1, do a DFS searching for a path of length 2 4.If there is no path of length 2, do a DFS searching for a path of length 3…

11. Iterative deepening search

15. Properties of iterative deepening search Complete? Yes Optimal? Yes, if step cost = 1 Time? (d+1)b 0 + d b 1 + (d-1)b 2 + … + b d = O(b d ) Space? O(bd)

16. Search with varying step costs BFS finds the path with the fewest steps, but does not always find the cheapest path

17. Uniform-cost search For each frontier node, save the total cost of the path from the initial state to that node Expand the frontier node with the lowest path cost Implementation: frontier is a priority queue ordered by path cost Equivalent to breadth-first if step costs all equal Equivalent to Dijkstra’s algorithm in general

18. Uniform-cost search example

19. Expansion order: (S,p,d,b,e,a,r,f,e,G)

20. Another example of uniform-cost search Source: WikipediaWikipedia

21. Properties of uniform-cost search Complete? Yes, if step cost is greater than some positive constant ε (we don’t want infinite sequences of steps that have a finite total cost) Optimal? Yes

22. Optimality of uniform-cost search Graph separation property: every path from the initial state to an unexplored state has to pass through a state on the frontier – Proved inductively Optimality of UCS: proof by contradiction – Suppose UCS terminates at goal state n with path cost g(n) but there exists another goal state n’ with g(n’) < g(n) – By the graph separation property, there must exist a node n” on the frontier that is on the optimal path to n’ – But because g(n”) ≤ g(n’) < g(n), n” should have been expanded first!

23. Properties of uniform-cost search Complete? Yes, if step cost is greater than some positive constant ε (we don’t want infinite sequences of steps that have a finite total cost) Optimal? Yes – nodes expanded in increasing order of path cost Time? Number of nodes with path cost ≤ cost of optimal solution (C*), O(b C*/ ε ) This can be greater than O(b d ): the search can explore long paths consisting of small steps before exploring shorter paths consisting of larger steps Space? O(b C*/ ε )

24. Review: Uninformed search strategies AlgorithmComplete?Optimal? Time complexity Space complexity BFS DFS IDS UCS b: maximum branching factor of the search tree d: depth of the optimal solution m: maximum length of any path in the state space C*: cost of optimal solution g(n): cost of path from start state to node n Yes No Yes If all step costs are equal Yes No O(b d ) O(b m ) O(b d ) O(bm) O(bd) Number of nodes with g(n) ≤ C*